Chap2Exercise_2

2. The yearly log returns on a stock are normally distributed with mean 0.1 and standard deviation 0.2. The stock is selling at $100 today. What is the probability that one year from now it is selling at $110 or more?

niter = 1e5
proba = rep(0,niter)
for (i in 1:niter)
  {
    r = rnorm(1,.1,.2)
    logPrice = log(100)+r
    proba[i] = as.numeric(logPrice>= log(110))
  }
mean(proba)

## [1] 0.51212

P(x>=110) = 1-P(x<110) = 1-cdf(110) As the log returns is normal IID, log return = log(P/P0) = log(110/100) = .09531018 now, i can use pnorm(log(110/100),.1,.2) This gives cdf(110). but we want P(x>=110) therefore 1-pnorm(log(110/100),.1,.2). Observe the following with the earlier mean(proba)

1-pnorm(log(110/100),.1,.2)

## [1] 0.509354